Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer

How many perfect squares divide $10^{10}$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

A perfect square $s$ divides $10^{10}$ if and only if $s=2^{a} \cdot 5^{b}$ where $a, b \in\{0,2,4,6,8,10\}$ . There are 36 choices, giving 36 different $s$ 's.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.